combinatorics has come of age. It had its beginnings in a number of puzzles which have still not lost their charm. Among these are EULER'S problem of the 36 officers and the KONIGSBERG bridge problem, BACHET's...
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ISBN:
(数字)9789401018265
ISBN:
(纸本)9789027705938
combinatorics has come of age. It had its beginnings in a number of puzzles which have still not lost their charm. Among these are EULER'S problem of the 36 officers and the KONIGSBERG bridge problem, BACHET's problem of the weights, and the Reverend T.P. KIRKMAN'S problem of the schoolgirls. Many of the topics treated in ROUSE BALL'S Recreational Mathe matics belong to combinatorial theory. All of this has now changed. The solution of the puzzles has led to a large and sophisticated theory with many complex ramifications. And it seems probable that the four color problem will only be solved in terms of as yet undiscovered deep results in graph theory. combinatorics and the theory of numbers have much in common. In both theories there are many prob lems which are easy to state in terms understandable by the layman, but whose solution depends on complicated and abstruse methods. And there are now interconnections between these theories in terms of which each enriches the other. combinatorics includes a diversity of topics which do however have interrelations in superficially unexpected ways. The instructional lectures included in these proceedings have been divided into six major areas: 1. Theory of designs; 2. Graph theory; 3. Combinatorial group theory; 4. Finite geometry; 5. Foundations, partitions and combinatorial geometry; 6. Coding theory. They are designed to give an overview of the classical foundations of the subjects treated and also some indication of the present frontiers of research.
There have been shifts toward more systematic and standardized methods for studying non-human primate facial signals, thanks to advancements like animalFACS. Additionally, there have been calls to better integrate the...
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There have been shifts toward more systematic and standardized methods for studying non-human primate facial signals, thanks to advancements like animalFACS. Additionally, there have been calls to better integrate the study of both facial and gestural communication in terms of theory and methodology. However, few studies have taken this important integrative step. By doing so, researchers could gain greater insight into how the physical flexibility of facial signals affects social flexibility. Our study combines both approaches to examine the relationship between the flexibility of physical form and the social function of chimpanzee facial "gestures". We used chimpFACS along with established gestural ethograms that provide insights into four key gesture properties and their associated variables documented in chimpanzee gestures. We specifically investigated how the combinatorics (i.e., the different combinations of facial muscle movements) and complexity (measured by the number of discrete facial muscle movements) of chimpanzee facial signals varied based on: (1) how many gesture variables they exhibit;(2) the presence of a specific goal;and (3) the context in which they were produced. Our findings indicate that facial signals produced with vocalizations exhibit fewer gesture variables, rarely align with specific goals, and exhibit reduced contextual flexibility. Furthermore, facial signals that include additional visual movements (such as those of the head) and other visual signals (like manual gestures) exhibit more gestural variables, are frequently aligned with specific goals, and exhibit greater contextual flexibility. Finally, we discovered that facial signals become more morphologically complex when they exhibit a greater number of gesture variables. Our findings indicate that facial "gesturing" significantly enhanced the facial signaling repertoire of chimpanzees, offering insights into the evolution of complex communication systems like human language.
This paper focuses on the degree of freedom and number of subdeterminants in a Pearson residual in a multiway contingency table. The results show that multidimensional residuals are represented as linear sum of determ...
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This paper focuses on the degree of freedom and number of subdeterminants in a Pearson residual in a multiway contingency table. The results show that multidimensional residuals are represented as linear sum of determinants of 2 x 2 submatrices, which can be viewed as information granules measuring the degree of statistical dependence. Geometrical interpretation of Pearson residual is investigated. Furthermore, the number of subdeterminants in a residual is equal to the degree of freedom in (2)-test statistic. Since the way of calculation of the number of subdeterminants corresponds to the construction of a statistical model for a contingency table, it has been found that the combinatorics of the number subdeterminants is closely related with permutation of attributes in a given table, where symmetric group may play an important role.
The action of a translation on a continuous object before its digitization generates several digital objects. This paper focuses on the combinatorics of the generated digital objects up to integer translations. In the...
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The action of a translation on a continuous object before its digitization generates several digital objects. This paper focuses on the combinatorics of the generated digital objects up to integer translations. In the general case, a worst-case upper bound is exhibited and proved to be reached on an example. Another upper bound is also proposed by making a link between the number of the digital objects and the boundary curve, through its self-intersections on the torus. An upper bound, quadratic in digital perimeter, is then derived in the convex case and eventually an asymptotic upper bound, quadratic in the grid resolution, is exhibited in the polygonal case. A few significant examples finish the paper.
We use some results about Betti numbers of coverings of complements of plane projective curves to discuss the problem of how combinatories determine the topology of line arrangement, finding a counterexample to a conj...
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We use some results about Betti numbers of coverings of complements of plane projective curves to discuss the problem of how combinatories determine the topology of line arrangement, finding a counterexample to a conjecture of Orlik.
A new bijection between the diagonally convex directed (dcd-) polyominoes and ternary trees makes it possible to enumerate the dcd-polyominoes according to several parameters (sources, diagonals, horizontal and vertic...
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A new bijection between the diagonally convex directed (dcd-) polyominoes and ternary trees makes it possible to enumerate the dcd-polyominoes according to several parameters (sources, diagonals, horizontal and vertical edges, target cells). For a part of these results we also give another proof, which is based on Raney's generalized lemma. Thanks to the fact that the diagonals of a dcd-polyomino can grow at most by one, the problem of q-enumeration of this object can be solved by an application of Gessel's q-analog of the Lagrange inversion formula.
In this paper we introduce a kind of directed graphs (digraphs) arranged in shifted rows of different lengths, which arise in a natural way related to problems of finding the number of certain families of canonical pr...
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In this paper we introduce a kind of directed graphs (digraphs) arranged in shifted rows of different lengths, which arise in a natural way related to problems of finding the number of certain families of canonical primitive connected cellular matrices of the p-Sylow (sic)(n) of GL(n)(q) formed by the upper unitriangular matrices over the finite field with q elements. Higman's conjecture states that the number of conjugacy classes of (sic)(n) is a polynomial in q. We associate a number, which we call the counter, to each directed graph, which gives additional information about the polynomial structure of the number of conjugacy classes. We focus on a family of digraphs, which we call parallelogramic digraphs, in which we have n rows of length k each one shifted one place to the right with respect to the previous one. We give explicit formulas for their counters for n up to 5. We prove also that the counters satisfy recurrence equations for fixed k when we vary n, proving thus a fact that was empirically observed by R.H. Harding and A.P. Heinz and proved by P. Sun for k up to 5. When n > 1, this number multiplied by (q - 1)(nk-1) corresponds to the cardinality of the family of canonical cellular nk x nk matrices over the field F-q with n pivot lines of length k and exactly k - 1 links connecting the pilots of the lines. We indicate other kinds of digraphs related to Higman's conjecture that establish lines of future research on this topic. (C) 2017 Elsevier Inc. All rights reserved.
The system of simple sequential processes with resources (S3PR) is a class of Petri nets that can model a typical class of resource allocation systems. This paper proposes a method to calculate the number of reachable...
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The system of simple sequential processes with resources (S3PR) is a class of Petri nets that can model a typical class of resource allocation systems. This paper proposes a method to calculate the number of reachable states of an S3PR. First of all, an upper bound of reachable states of an S3PR can be obtained by combinatorics. The number of unreachable states may be counted in the upper bound because of the existence of shared resources. Hence, the next step is to subtract the number of unreachable states from the upper bound. Here, siphons with two resources derived from resource circuits are used to obtain the number of the unreachable states. The resulting number after subtracting the number of unreachable states from the upper bound is what we are looking for. Finally, example calculations and analyses are conducted to verify the effectiveness of the method. (C) 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.
Guest editors Isabel Beichl and Francis Sullivan discuss the field of combinatorics in general and the articles in this special issue in particular, highlighting the recent upsurge in research in this area as well as ...
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Guest editors Isabel Beichl and Francis Sullivan discuss the field of combinatorics in general and the articles in this special issue in particular, highlighting the recent upsurge in research in this area as well as some interesting applications of it (such as in solving Sudoku puzzles).
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